Low-Dimensional Flows
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SET SYSTEM PARAMETERS
$$\begin{align}
\mathcal{L} &= \frac{1}{2}(f_*-a_1^{k_1})^2, \\
&= \frac{1}{2}(1-a_1^{1})^2,
\end{align}$$
$\dot{a}_i = -\frac{d \mathcal{L}}{d a_i}$.
LOSS AND PARAMETER FLOWS
loss $\mathcal{L}$
time $t$
parameters $a_i$
time $t$
THEORY COMPUTED BELOW THIS LINE
total order
$\ell = \sum_i k_i$
—
order prefactor
$\kappa = \prod_i k_i^{k_i/2}$
—
mean core parameter at init
$\beta = \frac{1}{d} \sum_i \frac{|a_i(0)|}{\sqrt{k_i}}$
—
shape parameters
$r_i = \frac{a_i(0)^2}{k_i \beta^2}$
—
shape integral
$F(\mathbf{r}) = \left(\frac{\ell}{2} - 1\right) \int_0^\infty \prod_i (s + r_i)^{-k_i/2} \, ds$
—
effective balanced init scale
$\beta_{\mathrm{eff}} = \begin{cases} \beta, & \text{if } \ell \leq 2 \\ F(\mathbf{r})^{-\frac{1}{\ell-2}} \beta, & \text{if } \ell > 2 \end{cases}$
—
rise time
$t_{\mathrm{rise}} = \begin{cases}
c^{-2}, & \text{if } \ell = 1, \\
-\frac{1}{\kappa c f_*} \cdot \log\left(\sqrt{\frac{\kappa c}{f_*}}\beta\right), & \text{if } \ell = 2, \\
\frac{1}{\ell - 2} \cdot \frac{1}{\kappa c f_*} \cdot \frac{F(\mathbf{r})}{\beta^{\ell-2}}. & \text{if } \ell > 2.
\end{cases}$
—